Negative Numbers

I. Introduction

Let's do a quick recap on how to subtract numbers on the number line? Suppose you are subtracting $3$ from $5$, you will start at 5 and take 3 steps to the left, like shown below. You will land up at $2$, and we already know that $5 - 3 = 2$.

Now what happens when you try to subtract a larger number from a smaller number? Let us say you are calculating $2 - 6$. Actually the number line shown above is incomplete. The real number line does not end at $0$ on the left. Like the right side, the left side is also a ray extends indefinitely, like shown below:

As you can see that as you move from $0$ towards the left, the numbers keep increasing in magnitude, but each number is shown with a negative (-) sign. These are $negative\ numbers$ . The numbers to the right of zero are positive numbers. When a number does not have a sign before it the number is always considered as positive. So, when we say $131$ we mean $+131$.

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Going back to our problem of calculating $2 - 6$. Like always, we start at $2$ and take $6$ steps towards the left, like below:

We have landed at the number $-4$. And that is correct. $2 - 6 = -4$

Now we will see how to perform common mathematical operations with negative numbers.

II. Comparing Numbers

It is easy to compare numbers on the number line. Any number to the left of another number is lesser than that number. If you look at the number line above, you can see that $-6$ is to the left of $-2$, therefore we can write $\unicode{x201C}-6 \lt -2\unicode{x201D}$. We can see that as we move towards the left the magnitude of number increases, but the number gets smaller and smaller. So, we can say that $\unicode{x201C}-1000 \lt -500\unicode{x201D}$ or in words $\unicode{x201C}negative\ one\ thousand\ is\ less\ than\ negative\ five\ hundred\unicode{x201D}$.

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Now let us try the following problem:

--------- Reference to question: e42605eb-bc53-4ee6-89ff-3f5f9de05262 ---------

III. Addition And Subtraction

It is easy to understand addition and subtraction of positive or negative numbers if you think of numbers having a direction and a magnitude. The negative numbers have a direction pointing to the left, whereas positive number have a direction pointing to the right. For example the number $-17$ has a direction to the left, and a magnitude of $17$, while the number $-11$ has a direction towards the right and a magnitude of $11$.

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Now we will understand the concept of addition and subtraction with the famous game of $\unicode{x201C}Tug\ Of\ War\unicode{x201D}$.

Let us imagine that there are $17$ people, all equally strong, pulling a rope towards the left, while $11$ people of the same strength pulling the rope towards the right. Which team do you think will win? And win by how much?

Since the left side or the negative side has $6$ more people than the right side or the positive side, we can say that the left wins by $6$. Mathematically we can write:

$11 - 17 = -6$ which is the same as $-17 + 11 = -6$

Now let us say we have two teams one having $8$ members and another having $14$ members, but this time both the teams are pulling towards the left. Although this may not be a very practical thing to do, but for the sake of understanding we will assume that both teams are pulling towards the left or the negative direction. So mathematically we can write this as $-8 - 14$.

So, we have two teams of sizes $8$ and $14$ pulling towards the left, and since there is nobody pulling towards the right, we will say that a team of size zero, is pulling towards the right.

Which side wins and by how much? The left side or the negative side wins by 22. So we can write: $-14 - 8 = -22$

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The things that you need to remember to easily handle addition or subtraction with negative numbers are as follows:

If both numbers are of the same sign, sum up their magnitude and the sign of the result is the sign of the two numbers. Look at the examples below:
$11 + 21 = 32$ (here both $11$ and $21$ are positive so we add the magnitudes)
$-21 - 15 = -36$ (here both $21$ and $15$ are negative, so we add the magnitudes and the sign of the result is same as the sign of the two numbers with is negative)
If both numbers are of different signs, we take the difference of the two magnitudes and the sign of the result is the sign of the larger number. For example:
$27 - 13 = 14$ (here the difference is $14$, and the sign of the larger number is positive so the answer is positive $14$
$12 - 25 = -13$ (here the difference is $13$ and the sign of the larger magnitude is negative, so the answer is $-13$)
$-31 + 22 = -9$ (here the difference is $9$ and the sign of the larger magnitude is negative, so the answer is $-9$)
$-24 + 46 = 22$ (here the difference is $22$ and the sign of the larger number is positive, so the answer is $22$)

Now let us take the example with several numbers together. How much is $-12 + 8 + 7 - 9 - 4 + 5$

We can solve this by performing the operations from left to right.

$-12 + 8 + 7 -9 - 4 + 5$, we know that $-12 + 8 = -4$

$= -4 + 7 - 9 - 4 + 5$, we know that $-4 + 7 = 3$

$= 3 - 9 - 4 + 5$, we know that $3 - 9 = - 6$

$= -6 - 4 + 5$, we know that $-6 - 4 = -10$

$= -10 + 5$, we know that $-10 + 5 = -5$

$= -5$

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IV. Multiplication & Division:

When you are performing multiplication or division it is easy to remember using same and opposite signs.

If both the operands have the same sign, the result is going to be positive.

If both the operands have different signs, the result is going to be negative.

We will use some example to understand this.

$2 \times 3 = 6$ here both $2$ and $3$ are positive, so the result is positive.

$2 \times (-3) = -6$, here $2$ is positive and $3$ is negative, so the result is negative.

$(-2) \times (-3) = 6$ here both $2$ and $3$ are negative, so the result is positive.

$(-2) \times 3 = -6$ here $2$ is negative and $3$ is positive, so the result is positive.

The same rules apply for division as well:

$12 \div 3 = 4$, here both $12$ and $3$ are positive, so the result is positive

$12 \div (-3) = -4$ here $12$ is positive and $3$ is negative, so the result is negative.

$(-12) \div (-3) = 4$ here both $12$ and $3$ are negative, so the result is positive.

$(-12) \div 3 = 4$ here $12$ is negative and 3 is positive, so the result is negative.

V. Handling Brackets And Signs:

How much is $-15 - (-7)$.

Whenever there is a sign before a bracket without a number, you can take the number outside the bracket as 1 and the operation as multiplication.

So, we can write:

$-15 - (-7)$

$= -15 - 1\times(-7)$

We know from our multiplication rules, that negative number multiplied by negative number gives us positive number.

So, when we open the bracket in our problem, we get $(-1) \times (-7)$ which gives us positive $7$.

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Our expression becomes:

$-15 + 7$

$= -8$